The Love of Numbers
Everyone knows, from an early age that ten x ten = 100. But, we can only write this result in that fashion because we are using a base ten number system.
Suppose we used a base 8 number system (I am sure we would have developed a base 8 number system IF we had 8 fingers on our two hands, we have a base ten system precisely because we have ten fingers on our two hands).
In a base 8 number system the digits used would be 0, 1, 2, 3, 4, 5, 6, 7, only
To write the number “eight” in base 8 we use “10”, which means one of our base (8) and no units.
Thus the first 40 counting numbers in base eight would be:
1, 2, 3, 4, 5, 6, 7,1 0, 11, 12, 13,1 4, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 2 27, 30, 31, 32, 33, 34, 35,
36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 50
Therefore, in base eight we write ten x ten as 144 (64+32+4)
Other base systems have many uses, for instance all computers are programmed in binary code, ie numbers written in base two. You may think that base two would limit your options, but look at all the wonderful things that computers can do, by just utilising a code system where the only two digits are 0 or 1. Thus ten in base 2 is written as 1010. In base two: ten x ten = 110010
To write the whole sum in base two we write 1010 x1010 = 110010
Some of you may recall that I proved a very interesting result about Pythagoras’ Theorem using numbers in bases 3, 4, and 5. (The result: if a, b, c, are all integers and then the product abc is exactly divisible by 60). This result is very difficult to prove just using base ten.